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Properties

Label 112.16.a.a.1.1
Level $112$
Weight $16$
Character 112.1
Self dual yes
Analytic conductor $159.817$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,16,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(159.816725712\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 112.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1350.00 q^{3} -81060.0 q^{5} -823543. q^{7} -1.25264e7 q^{9} -7.01212e7 q^{11} +1.51470e8 q^{13} +1.09431e8 q^{15} -2.49757e8 q^{17} +6.47686e9 q^{19} +1.11178e9 q^{21} +2.11292e10 q^{23} -2.39469e10 q^{25} +3.62817e10 q^{27} +7.79483e9 q^{29} +9.50321e10 q^{31} +9.46636e10 q^{33} +6.67564e10 q^{35} -8.70082e11 q^{37} -2.04484e11 q^{39} +1.00767e12 q^{41} -1.55008e11 q^{43} +1.01539e12 q^{45} +2.55197e12 q^{47} +6.78223e11 q^{49} +3.37171e11 q^{51} +4.04765e12 q^{53} +5.68402e12 q^{55} -8.74376e12 q^{57} +1.25992e13 q^{59} -3.99250e13 q^{61} +1.03160e13 q^{63} -1.22781e13 q^{65} +4.84238e13 q^{67} -2.85244e13 q^{69} -3.76931e13 q^{71} +1.41416e14 q^{73} +3.23283e13 q^{75} +5.77478e13 q^{77} -2.47021e14 q^{79} +1.30760e14 q^{81} -2.78879e12 q^{83} +2.02453e13 q^{85} -1.05230e13 q^{87} -5.83963e12 q^{89} -1.24742e14 q^{91} -1.28293e14 q^{93} -5.25014e14 q^{95} +2.78027e14 q^{97} +8.78366e14 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1350.00 −0.356389 −0.178195 − 0.983995i \(-0.557026\pi\)
−0.178195 + 0.983995i \(0.557026\pi\)
\(4\) 0 0
\(5\) −81060.0 −0.464015 −0.232007 − 0.972714i \(-0.574529\pi\)
−0.232007 + 0.972714i \(0.574529\pi\)
\(6\) 0 0
\(7\) −823543. −0.377964
\(8\) 0 0
\(9\) −1.25264e7 −0.872987
\(10\) 0 0
\(11\) −7.01212e7 −1.08494 −0.542468 − 0.840076i \(-0.682510\pi\)
−0.542468 + 0.840076i \(0.682510\pi\)
\(12\) 0 0
\(13\) 1.51470e8 0.669499 0.334750 − 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(14\) 0 0
\(15\) 1.09431e8 0.165370
\(16\) 0 0
\(17\) −2.49757e8 −0.147622 −0.0738108 − 0.997272i \(-0.523516\pi\)
−0.0738108 + 0.997272i \(0.523516\pi\)
\(18\) 0 0
\(19\) 6.47686e9 1.66231 0.831155 − 0.556040i \(-0.187680\pi\)
0.831155 + 0.556040i \(0.187680\pi\)
\(20\) 0 0
\(21\) 1.11178e9 0.134702
\(22\) 0 0
\(23\) 2.11292e10 1.29397 0.646985 − 0.762503i \(-0.276030\pi\)
0.646985 + 0.762503i \(0.276030\pi\)
\(24\) 0 0
\(25\) −2.39469e10 −0.784691
\(26\) 0 0
\(27\) 3.62817e10 0.667512
\(28\) 0 0
\(29\) 7.79483e9 0.0839115 0.0419557 − 0.999119i \(-0.486641\pi\)
0.0419557 + 0.999119i \(0.486641\pi\)
\(30\) 0 0
\(31\) 9.50321e10 0.620379 0.310190 − 0.950675i \(-0.399607\pi\)
0.310190 + 0.950675i \(0.399607\pi\)
\(32\) 0 0
\(33\) 9.46636e10 0.386659
\(34\) 0 0
\(35\) 6.67564e10 0.175381
\(36\) 0 0
\(37\) −8.70082e11 −1.50677 −0.753386 − 0.657579i \(-0.771581\pi\)
−0.753386 + 0.657579i \(0.771581\pi\)
\(38\) 0 0
\(39\) −2.04484e11 −0.238602
\(40\) 0 0
\(41\) 1.00767e12 0.808049 0.404025 − 0.914748i \(-0.367611\pi\)
0.404025 + 0.914748i \(0.367611\pi\)
\(42\) 0 0
\(43\) −1.55008e11 −0.0869640 −0.0434820 − 0.999054i \(-0.513845\pi\)
−0.0434820 + 0.999054i \(0.513845\pi\)
\(44\) 0 0
\(45\) 1.01539e12 0.405079
\(46\) 0 0
\(47\) 2.55197e12 0.734753 0.367377 − 0.930072i \(-0.380256\pi\)
0.367377 + 0.930072i \(0.380256\pi\)
\(48\) 0 0
\(49\) 6.78223e11 0.142857
\(50\) 0 0
\(51\) 3.37171e11 0.0526107
\(52\) 0 0
\(53\) 4.04765e12 0.473297 0.236648 − 0.971595i \(-0.423951\pi\)
0.236648 + 0.971595i \(0.423951\pi\)
\(54\) 0 0
\(55\) 5.68402e12 0.503426
\(56\) 0 0
\(57\) −8.74376e12 −0.592429
\(58\) 0 0
\(59\) 1.25992e13 0.659105 0.329552 − 0.944137i \(-0.393102\pi\)
0.329552 + 0.944137i \(0.393102\pi\)
\(60\) 0 0
\(61\) −3.99250e13 −1.62657 −0.813283 − 0.581869i \(-0.802322\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(62\) 0 0
\(63\) 1.03160e13 0.329958
\(64\) 0 0
\(65\) −1.22781e13 −0.310657
\(66\) 0 0
\(67\) 4.84238e13 0.976108 0.488054 − 0.872814i \(-0.337707\pi\)
0.488054 + 0.872814i \(0.337707\pi\)
\(68\) 0 0
\(69\) −2.85244e13 −0.461157
\(70\) 0 0
\(71\) −3.76931e13 −0.491840 −0.245920 − 0.969290i \(-0.579090\pi\)
−0.245920 + 0.969290i \(0.579090\pi\)
\(72\) 0 0
\(73\) 1.41416e14 1.49823 0.749114 − 0.662442i \(-0.230480\pi\)
0.749114 + 0.662442i \(0.230480\pi\)
\(74\) 0 0
\(75\) 3.23283e13 0.279655
\(76\) 0 0
\(77\) 5.77478e13 0.410067
\(78\) 0 0
\(79\) −2.47021e14 −1.44720 −0.723602 − 0.690217i \(-0.757515\pi\)
−0.723602 + 0.690217i \(0.757515\pi\)
\(80\) 0 0
\(81\) 1.30760e14 0.635093
\(82\) 0 0
\(83\) −2.78879e12 −0.0112805 −0.00564027 − 0.999984i \(-0.501795\pi\)
−0.00564027 + 0.999984i \(0.501795\pi\)
\(84\) 0 0
\(85\) 2.02453e13 0.0684986
\(86\) 0 0
\(87\) −1.05230e13 −0.0299051
\(88\) 0 0
\(89\) −5.83963e12 −0.0139946 −0.00699730 − 0.999976i \(-0.502227\pi\)
−0.00699730 + 0.999976i \(0.502227\pi\)
\(90\) 0 0
\(91\) −1.24742e14 −0.253047
\(92\) 0 0
\(93\) −1.28293e14 −0.221096
\(94\) 0 0
\(95\) −5.25014e14 −0.771336
\(96\) 0 0
\(97\) 2.78027e14 0.349381 0.174690 − 0.984623i \(-0.444108\pi\)
0.174690 + 0.984623i \(0.444108\pi\)
\(98\) 0 0
\(99\) 8.78366e14 0.947135
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 112.16.a.a.1.1 1
4.3 odd 2 14.16.a.a.1.1 ✓ 1
12.11 even 2 126.16.a.e.1.1 1
28.3 even 6 98.16.c.c.79.1 2
28.11 odd 6 98.16.c.b.79.1 2
28.19 even 6 98.16.c.c.67.1 2
28.23 odd 6 98.16.c.b.67.1 2
28.27 even 2 98.16.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.16.a.a.1.1 ✓ 1 4.3 odd 2
98.16.a.b.1.1 1 28.27 even 2
98.16.c.b.67.1 2 28.23 odd 6
98.16.c.b.79.1 2 28.11 odd 6
98.16.c.c.67.1 2 28.19 even 6
98.16.c.c.79.1 2 28.3 even 6
112.16.a.a.1.1 1 1.1 even 1 trivial
126.16.a.e.1.1 1 12.11 even 2